Torsion of a multilayer base with elastic connections between layers by cylindrical punch with a flat sole was considered. The Hankel integral transform of the first order and the method of the compliance functions were used. Two auxiliary functions connected with transformants of tangential stresses and displacements points of the upper boundary of the layer were introduced for each layer. The components of the stress-strain state were represented as a linear combination of these functions. The recurrent formulas binding auxiliary functions of neighboring layers were built based on conditions of joint deformation of neighboring layers. The compliance functions were introduced. The recurrent formulas binding the compliance functions of neighboring layers were built. The problem was resolved into the integral equation. The kernel of the integral equation contains Sonine-Weber integral. The approximate solution of equation was found by the method of mechanical quadratures. Mechanical effects for one-layer and two-layer bases were obtained. Elastic connections between the layers of base led to a reduction of contact stresses in comparison with the case of full contact. Decrease of the shear modulus of one of the layers of a two-layer base was reduced to decrease of the contact stresses, as in the cause of ideal contact and when elastic connections are between the layers of the base.
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